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NCERT Solutions

Sets

Madhya Pradesh Board · Class 11 · Mathematics

NCERT Solutions for Sets — Madhya Pradesh Board Class 11 Mathematics.

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EXERCISE 1.1

1Which of the following are sets ? Justify your answer.Show solution
- (i) Set — the months beginning with J are well-defined: January, June, July.
- (ii) Not a set — “ten most talented” is not well-defined, since the criterion may differ from person to person.
- (iii) Not a set — “best” cricket batsmen is not a well-defined collection.
- (iv) Set — the boys in your class form a well-defined collection.
- (v) Set — natural numbers less than 100 are well-defined.
- (vi) Set — novels written by Munshi Prem Chand form a well-defined collection.
- (vii) Set — even integers are well-defined.
- (viii) Set — the questions in this chapter are well-defined.
- (ix) Not a set — “most dangerous animals” is not a well-defined collection.

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2Let A={1,2,3,4,5,6}A = \{1, 2, 3, 4, 5, 6\}. Insert the appropriate symbol ∈\in or ∉\notin in the blank spaces:Show solution
For A={1,2,3,4,5,6}A=\{1,2,3,4,5,6\}:
- 55 is an element of AA, so 5∈A5 \in A
- 88 is not in AA, so 8∉A8 \notin A
- 00 is not in AA, so 0∉A0 \notin A
- 44 is in AA, so 4∈A4 \in A
- 22 is in AA, so 2∈A2 \in A
- 1010 is not in AA, so 10∉A10 \notin A

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3Write the following sets in roster form:Show solution
Write each set by listing all distinct elements:
- (i) Integers from −3-3 to 66
- (ii) Natural numbers less than 66
- (iii) Two-digit natural numbers whose digits add to 88: 17,26,35,44,53,62,71,80,8917,26,35,44,53,62,71,80,89
- (iv) Prime divisors of 6060: 2,3,52,3,5
- (v) Letters in TRIGONOMETRY without repetition: {T,R,I,G,O,N,M,E,Y}\{T,R,I,G,O,N,M,E,Y\}
- (vi) Letters in BETTER without repetition: {B,E,T,R}\{B,E,T,R\}

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4Write the following sets in the set-builder form :Show solution
Describe each roster set by its common property:
- (i) 3,6,9,123,6,9,12 are multiples of 33.
- (ii) 2,4,8,16,322,4,8,16,32 are powers of 22 from 212^1 to 252^5.
- (iii) 5,25,125,6255,25,125,625 are powers of 55 from 515^1 to 545^4.
- (iv) 2,4,6,…2,4,6,\dots are even natural numbers.
- (v) 1,4,9,…,1001,4,9,\dots,100 are perfect squares from 121^2 to 10210^2.

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5List all the elements of the following sets :Show solution
List the elements satisfying each description:
- (i) odd natural numbers: {1,3,5,7,9,… }\{1,3,5,7,9,\dots\}
- (ii) integers between −12-\tfrac12 and 92\tfrac92: −1,0,1,2,3,4-1,0,1,2,3,4
- (iii) integers with x2≤4x^2\le 4: −2,−1,0,1,2-2,-1,0,1,2
- (iv) distinct letters in LOYAL: {L,O,Y,A}\{L,O,Y,A\}
- (v) months without 31 days: January has 31, so the months are February, April, June, September, November
- (vi) consonants in alphabet that precede kk: the consonants before kk are {b,c,d,f,g,h,j}\{b,c,d,f,g,h,j\} if read directly from the alphabet; but since the textbook asks the set-builder description, the elements are those consonants. In roster form this is {b,c,d,f,g,h,j}\{b,c,d,f,g,h,j\}.

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6Match each of the set on the left in the roster form with the same set on the right described in set-builder form:Show solution
Match by finding the same elements:
- (i) PRINCIPAL has letters {P,R,I,N,C,A,L}\{P,R,I,N,C,A,L\}, matching (d).
- (ii) {0}\{0\} matches (c) because x+1=1⇒x=0x+1=1 \Rightarrow x=0.
- (iii) {1,2,3,6,9,18}\{1,2,3,6,9,18\} are the positive divisors of 1818, matching (a).
- (iv) x2−9=0⇒x=3,−3x^2-9=0 \Rightarrow x=3,-3, matching (b).

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EXERCISE 1.2

1Which of the following are examples of the null setShow solution
- (i) The set of odd natural numbers divisible by 2 is empty, since an odd number is never divisible by 2.
- (ii) The set of even prime numbers is not empty, because 22 is an even prime number.
- (iii) x<5x<5 and x>7x>7 cannot happen together, so the set is empty.
- (iv) Two parallel lines do not meet, so there is no common point; hence the set is empty.

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2Which of the following sets are finite or infiniteShow solution
- (i) Months of a year: finite.
- (ii) {1,2,3,… }\{1,2,3,\dots\}: infinite.
- (iii) {1,2,3,…,99,100}\{1,2,3,\dots,99,100\}: finite.
- (iv) Positive integers greater than 100: infinite.
- (v) Prime numbers less than 99: finite.

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3State whether each of the following set is finite or infinite:Show solution
- (i) Lines parallel to the xx-axis: infinite.
- (ii) Letters in the English alphabet: finite.
- (iii) Multiples of 5: infinite.
- (iv) Animals living on the earth: infinite.
- (v) Circles passing through the origin: infinite.

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4In the following, state whether A = B or not:Show solution
- (i) Both sets have exactly the same elements, so A=BA=B.
- (ii) A={4,8,12,16}A=\{4,8,12,16\} but 18∈B18\in B and 18∉A18\notin A, so A≠BA\ne B.
- (iii) The set of positive even integers ≤10\le 10 is {2,4,6,8,10}\{2,4,6,8,10\}, so A=BA=B.
- (iv) AA is the set of multiples of 10, while B={10,15,20,25,30,… }B=\{10,15,20,25,30,\dots\} includes numbers like 15 and 25 that are not multiples of 10, so A≠BA\ne B.

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5Are the following pair of sets equal? Give reasons.Show solution
- (i) Equal. The letters in ALLOY are {A,L,O,Y}\{A,L,O,Y\} and in LOYAL are also {L,O,Y,A}\{L,O,Y,A\}; repetition and order do not matter.
- (ii) Not equal. A={2,3}A=\{2,3\}, but solving x2+5x+6=0x^2+5x+6=0 gives (x+2)(x+3)=0(x+2)(x+3)=0, so the solution set is {−2,−3}\{-2,-3\}, which is different from AA.

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6From the sets given below, select equal sets:Show solution
- E={−1,1}E=\{-1,1\} and G={1,−1}G=\{1,-1\}, so they are equal sets.
- F={0,a}F=\{0,a\} and H={0,1}H=\{0,1\} are not equal because they do not have the same elements.
- The other sets are not equal to these or to each other by their elements.

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EXERCISE 1.3

1Make correct statements by filling in the symbols ⊂\subset or ⊄\not\subset in the blank spaces :Show solution
Check whether every element of the first set belongs to the second:
- (i) {2,3,4}⊂{1,2,3,4,5}\{2,3,4\}\subset\{1,2,3,4,5\}.
- (ii) {a,b,c}⊄{b,c,d}\{a,b,c\}\not\subset\{b,c,d\} because aa is missing.
- (iii) All Class XI students of your school are students of your school, so subset.
- (iv) Every circle in a plane is not necessarily of radius 1, so not a subset.
- (v) A triangle is not a rectangle, so not a subset.
- (vi) Every equilateral triangle is a triangle, so subset.
- (vii) Every even natural number is an integer, so subset.

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2Examine whether the following statements are true or false:Show solution
- (i) {a,b}⊂{b,c,a}\{a,b\}\subset\{b,c,a\}: true.
- (ii) {a,c}\{a,c\} is not a subset of vowels, because cc is not a vowel.
- (iii) {1,2,3}⊂{1,3,5}\{1,2,3\}\subset\{1,3,5\} is false because 22 is missing.
- (iv) {a}⊂{a,b,c}\{a\}\subset\{a,b,c\} is true.
- (v) {a}∈{a,b,c}\{a\}\in\{a,b,c\} is false because the elements are letters, not the set {a}\{a\}.
- (vi) Even natural numbers less than 6 are {2,4}\{2,4\}, and both divide 36, so true.

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3Let A={1,2,{3,4},5}A = \{ 1, 2, \{ 3, 4 \}, 5 \}. Which of the following statements are incorrect and why?Show solution
The incorrect statements are:
- (i) {3,4}⊂A\{3,4\}\subset A is incorrect because A={1,2,{3,4},5}A=\{1,2,\{3,4\},5\} contains the set {3,4}\{3,4\} as an element, not the numbers 3 and 4 as separate elements.
- (v) 1⊂A1\subset A is incorrect because 1 is an element, not a set; only sets can be subsets.
- (vii) {1,2,5}∈A\{1,2,5\}\in A is incorrect because AA contains 1, 2, {3,4}\{3,4\}, and 5, but not the set {1,2,5}\{1,2,5\} as an element.
- (viii) {1,2,3}⊂A\{1,2,3\}\subset A is incorrect because 3 is not an element of AA.
- (ix) ϕ∈A\phi\in A is incorrect because the empty set is not listed as an element of AA.
- (xi) {ϕ}⊂A\{\phi\}\subset A is incorrect because ϕ\phi is not an element of AA.

The correct ones are (ii), (iii), (iv), (vi), (x).

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4Write down all the subsets of the following setsShow solution
List all possible subsets:
- (i) For {a}\{a\}: ∅\emptyset and {a}\{a\}.
- (ii) For {a,b}\{a,b\}: ∅,{a},{b},{a,b}\emptyset, \{a\}, \{b\}, \{a,b\}.
- (iii) For {1,2,3}\{1,2,3\}: all 23=82^3=8 subsets.
- (iv) The empty set has only one subset, itself.

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5Write the following as intervals :Show solution
Convert each set-builder description to interval notation:
- (i) −4<x≤6-4<x\le 6 gives (−4,6](-4,6]
- (ii) −12<x<−10-12<x<-10 gives (−12,−10)(-12,-10)
- (iii) 0≤x<70\le x<7 gives [0,7)[0,7)
- (iv) 3≤x≤43\le x\le 4 gives [3,4][3,4]

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6Write the following intervals in set-builder form :Show solution
Use the interval endpoints and whether they are included:
- (−3,0)( -3,0 ) means −3<x<0-3<x<0
- [6,12][6,12] means 6≤x≤126\le x\le 12
- (6,12](6,12] means 6<x≤126<x\le 12
- [−23,5)[-23,5) means −23≤x<5-23\le x<5

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7What universal set(s) would you propose for each of the following :Show solution
A universal set should be a larger set containing the given set as a subset.
- (i) For right triangles, a suitable universal set is the set of all triangles in a plane.
- (ii) For isosceles triangles, a suitable universal set is also the set of all triangles in the same plane.

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8Given the sets A={1,3,5}A = \{1, 3, 5\}, B={2,4,6}B = \{2, 4, 6\} and C={0,2,4,6,8}C = \{0, 2, 4, 6, 8\}, which of the following may be considered as universal set (s) for all the three sets AA, BB and CCShow solution
A universal set must contain A, B, and C.
- (i) {0,1,2,3,4,5,6}\{0,1,2,3,4,5,6\} contains all elements of AA, BB, and CC.
- (ii) ϕ\phi cannot be a universal set.
- (iii) {0,1,2,3,4,5,6,7,8,9,10}\{0,1,2,3,4,5,6,7,8,9,10\} contains all three sets.
- (iv) {1,2,3,4,5,6,7,8}\{1,2,3,4,5,6,7,8\} does not contain 00, so it cannot contain CC.
Therefore, the possible universal sets are (i) and (iii).

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EXERCISE 1.4

1Find the union of each of the following pairs of sets :Show solution
Union means collecting all elements from both sets without repetition:
- (i) {1,3,5}∪{1,2,3}={1,2,3,5}\{1,3,5\}\cup\{1,2,3\}=\{1,2,3,5\}
- (ii) {a,e,i,o,u}∪{a,b,c}={a,b,c,e,i,o,u}\{a,e,i,o,u\}\cup\{a,b,c\}=\{a,b,c,e,i,o,u\}
- (iii) multiples of 3 union natural numbers less than 6 gives {1,2,3,4,5}\{1,2,3,4,5\}
- (iv) {2,3,4,5,6}∪{7,8,9}={2,3,4,5,6,7,8,9}\{2,3,4,5,6\}\cup\{7,8,9\}=\{2,3,4,5,6,7,8,9\}
- (v) with empty set, union is the set itself.

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2Let A={a,b}A = \{ a, b \}, B={a,b,c}B = \{ a, b, c \}. Is A⊂BA \subset B? What is A∪BA \cup B?Show solution
A={a,b}A=\{a,b\} and B={a,b,c}B=\{a,b,c\}.
Every element of AA is in BB, so **A⊂BA\subset B**.
The union contains all distinct elements of both sets, so
A∪B={a,b,c}A\cup B=\{a,b,c\}.

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3If AA and BB are two sets such that A⊂BA \subset B, then what is A∪BA \cup B?Show solution
If A⊂BA\subset B, then every element of AA is already in BB. So the union adds nothing new, and
A∪B=B.A\cup B = B.

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4If A={1,2,3,4}A = \{ 1, 2, 3, 4 \}, B={3,4,5,6}B = \{ 3, 4, 5, 6 \}, C={5,6,7,8}C = \{ 5, 6, 7, 8 \} and D={7,8,9,10}D = \{ 7, 8, 9, 10 \}; findShow solution
Find each union by combining all distinct elements:
- A∪B={1,2,3,4,5,6}A\cup B=\{1,2,3,4,5,6\}
- A∪C={1,2,3,4,5,6,7,8}A\cup C=\{1,2,3,4,5,6,7,8\}
- B∪C={2,3,4,5,6,7,8}B\cup C=\{2,3,4,5,6,7,8\}
- B∪D={2,3,4,5,6,7,8,9,10}B\cup D=\{2,3,4,5,6,7,8,9,10\}
- A∪B∪C={1,2,3,4,5,6,7,8}A\cup B\cup C=\{1,2,3,4,5,6,7,8\}
- A∪B∪D={1,2,3,4,5,6,7,8,9,10}A\cup B\cup D=\{1,2,3,4,5,6,7,8,9,10\}
- B∪C∪D={2,3,4,5,6,7,8,9,10}B\cup C\cup D=\{2,3,4,5,6,7,8,9,10\}

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5Find the intersection of each pair of sets of question 1 above.Show solution
Find common elements of each pair from Question 1:
- (i) {1,3,5}∩{1,2,3}={1,3}\{1,3,5\}\cap\{1,2,3\}=\{1,3\}
- (ii) {a,e,i,o,u}∩{a,b,c}={a}\{a,e,i,o,u\}\cap\{a,b,c\}=\{a\}
- (iii) Multiples of 3 and numbers less than 6: common elements are {3}\{3\}.
- (iv) {2,3,4,5,6}∩{7,8,9}=∅\{2,3,4,5,6\}\cap\{7,8,9\}=\emptyset
- (v) Any set intersect empty set is empty.
- (vi) Both are empty set on intersection if one is empty.
- (vii) Even natural numbers and natural numbers 1<x≤61<x\le 6 have common elements {2,4,6}\{2,4,6\} if using the given actual sets.

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6If A={3,5,7,9,11}A = \{ 3, 5, 7, 9, 11 \}, B={7,9,11,13}B = \{ 7, 9, 11, 13 \}, C={11,13,15}C = \{ 11, 13, 15 \} and D={15,17}D = \{ 15, 17 \}; find
7If A={x:x is a natural number}A = \{x : x \text{ is a natural number}\}, B={x:x is an even natural number}B = \{x : x \text{ is an even natural number}\}
C={x:x is an odd natural number}C = \{x : x \text{ is an odd natural number}\} and D={x:x is a prime number}D = \{x : x \text{ is a prime number}\}, find
8Which of the following pairs of sets are disjoint
9If A={3,6,9,12,15,18,21}A = \{3, 6, 9, 12, 15, 18, 21\}, B={4,8,12,16,20}B = \{4, 8, 12, 16, 20\},
C={2,4,6,8,10,12,14,16}C = \{2, 4, 6, 8, 10, 12, 14, 16\}, D={5,10,15,20}D = \{5, 10, 15, 20\}; find
10If X={a,b,c,d}X = \{a, b, c, d\} and Y={f,b,d,g}Y = \{f, b, d, g\}, find
11If RR is the set of real numbers and QQ is the set of rational numbers, then what is R−QR - Q?
12State whether each of the following statement is true or false. Justify your answer.

EXERCISE 1.5

1Let U={1,2,3,4,5,6,7,8,9}U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}, A={1,2,3,4}A = \{1, 2, 3, 4\}, B={2,4,6,8}B = \{2, 4, 6, 8\} and C={3,4,5,6}C = \{3, 4, 5, 6\}. Find
2If U={a,b,c,d,e,f,g,h}U = \{a, b, c, d, e, f, g, h\}, find the complements of the following sets :
3Taking the set of natural numbers as the universal set, write down the complements of the following sets:
4If U={1,2,3,4,5,6,7,8,9}U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}, A={2,4,6,8}A = \{2, 4, 6, 8\} and B={2,3,5,7}B = \{2, 3, 5, 7\}. Verify that
5Draw appropriate Venn diagram for each of the following :
6Let UU be the set of all triangles in a plane. If AA is the set of all triangles with at least one angle different from 60∘60^\circ, what is A′A'?
7Fill in the blanks to make each of the following a true statement :

Miscellaneous Exercise on Chapter 1

1Decide, among the following sets, which sets are subsets of one and another:
2In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.
3Let A, B, and C be the sets such that A∪B=A∪CA \cup B = A \cup C and A∩B=A∩CA \cap B = A \cap C. Show that B=CB = C.
4Show that the following four conditions are equivalent:
5Show that if A⊂BA \subset B, then C−B⊂C−AC - B \subset C - A.
6Show that for any sets A and B,
A=(A∩B)∪(A−B)A = (A \cap B) \cup (A - B) and A∪(B−A)=(A∪B)A \cup (B - A) = (A \cup B)
7Using properties of sets, show that
8Show that A∩B=A∩CA \cap B = A \cap C need not imply B=CB = C.
9Let A and B be sets. If A∩X=B∩X=ϕA \cap X = B \cap X = \phi and A∪X=B∪XA \cup X = B \cup X for some set X, show that A=BA = B.
10Find sets A, B and C such that A∩BA \cap B, B∩CB \cap C and A∩CA \cap C are non-empty sets and A∩B∩C=ϕA \cap B \cap C = \phi.

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Frequently Asked Questions

What are the important topics in Sets for Madhya Pradesh Board Class 11 Mathematics?
Key topics in Sets include How to Find Complement, Union, Intersection, and Difference — Step-by-Step Decision Flowchart, Chapter Overview — Sets and All Key Concepts, Chapter 1: Sets — Complete Concept Map. These are the concepts Madhya Pradesh Board Class 11 examiners draw on most — study them first, then practise related questions.
How to score full marks in Sets — Madhya Pradesh Board Class 11 Mathematics?
Understand the core concepts first, then work through the 146 practice questions available for this chapter. Revise formulas and definitions regularly, and use flashcards for quick recall before the exam.
Where can I get free NCERT Solutions for Sets Class 11 Mathematics?
This page has free step-by-step NCERT Solutions for every exercise question in Sets (Madhya Pradesh Board Class 11 Mathematics) — written the way examiners award marks: given, formula, working, answer.

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